A Basic Result on the Superposition of Arrival Processes in Deterministic Networks
aa r X i v : . [ c s . PF ] S e p A Basic Result on the Superposition of ArrivalProcesses in Deterministic Networks
Yuming JiangNTNU, Norwegian University of Science and Technology, Norway
Abstract —Time-Sensitive Networking (TSN) and DeterministicNetworking (DetNet) are emerging standards to enable deter-ministic, delay-critical communication in such networks. Thisnaturally (re-)calls attention to the network calculus theory(NC), since a rich set of results for delay guarantee analysishave already been developed there. One could anticipate animmediate adoption of those existing network calculus results toTSN and DetNet. However, the fundamental difference betweenthe traffic specification adopted in TSN and DetNet and thosetraffic models in NC makes this difficult, let alone that thereis a long-standing open challenge in NC. To address them, thispaper considers an arrival time function based max-plus NCtraffic model. In particular, a relationship between the TSN /DetNet traffic specification and the NC traffic model is proved. Inaddition, the superposition property of the arrival time functionbased NC traffic model is found and proved. Appealingly, theproved superposition property shows a clear analogy with thatof a well-known counterpart traffic model in NC. These resultshelp make an important step towards the development of a systemtheory for delay guarantee analysis of TSN / DetNet networks.
Index Terms —Time-Sensitive Networking (TSN); DeterministicNetworking (DetNet); Network Calculus; Max-Plus NetworkCalculus; Superposition Property; Max-Plus Arrival Curve
I. I
NTRODUCTION
Time-sensitive applications can be broadly found in in-dustrial process control, machine control and live stream-ing of audio and video. To support such applications andto enable deterministic delay-critical communication, Time-Sensitive Networking (TSN) and Deterministic Networking(DetNet) are emerging standards respectively introduced bythe IEEE TSN Task Group [1] for Layer 2 Ethernet switches,and by the IETF DetNet Working Group [2] for more generalnetwork settings. In both TSN and DetNet, the traffic speci-fication (TSpec) uses two parameters to model a flow or anarrival process: a time interval and the maximum number ofpackets in the interval [1][2].Since a rich set of results for delay guarantee analysis havealready been developed in the network calculus theory (NC),e.g. [3][4][5][6][7][8][9][10][11][12], one could anticipate animmediate adoption of those existing NC results to TSN andDetNet. However, the traffic specification adopted in TSN andDetNet is fundamentally different from those traffic models inNC, which makes the adoption difficult, let alone that thereis a long-standing open challenge in the related part of NC.To address this difficulty and the open challenge forms themotivation and objective of the present paper.Specifically, in this paper, a packet arrival time functionbased traffic model related to the max-plus branch of NC[5][9][10][11][12] is introduced. We prove that there is a mapping between the TSN / DetNet traffic specification andthe max-plus traffic model, which establishes an important linkfor making use of NC results to TSN / DetNet analysis.However, for this max-plus traffic model, there is a long-standing problem, which is its superposition property, i.e.,the aggregate arrival process resulted from the aggregationof multiple arrival processes can be characterized using thesame model as for the individual arrival processes. Specifically,the superposition property has surprisingly not been found orproved directly on the model itself for long time due to aninherent challenge [5][9][10][11][12]. Since the superpositionproperty is one of the most basic properties needed for networkperformance analysis [7] [8], this calls for an urgent need ofinvestigation.The inherent challenge [10][12] is due to the complexformulation of the arrival time function of the aggregateprocess in terms of the arrival time functions of the individualarrival processes, making it difficult (if not impossible) tocharacterize the aggregate process directly on this aggregatearrival time function. To bypass this challenge, an indirectapproach has been considered in the literature [5][10]. How-ever, this indirect approach requires packet length information[5][10][12], which is not available or needed in the arrivaltime function description of the arrival process as is the casein TSN and DetNet.In this paper, a novel approach is used which works directlyon the arrival time functions, fundamentally different fromthe indirect approach. Based on this direct approach, thesuperposition property of the arrival time function based max-plus traffic model is found and proved. Appealingly, theproved superposition property has a clear analogy with theaggregation property of the well-known ( σ, ρ ) traffic model[3] in the min-plus branch of NC [3][6], and is (much)better than that from the indirect approach. The superpositionproperty and its proof using the direct approach form anothercontribution that is crucial to both NC and the future use ofNC results to TSN and Det Net.The rest is organized as follows. In Sec. II, the max-plus traffic model is introduced, together with the proof ofthe mapping between it and the TSN / DetNet TSpec. InSec. III, the inherent challenge is first discussed, followed bythe superposition property with detailed proof. In Sec. IV, acomparison study of results using the indirect approach andthe direct approach is provided. This comparison implies theimportance of the superposition property proved in this paper.Finally, concluding remarks are made in Sec. V.I. T HE M AX -P LUS T RAFFIC M ODEL AND T HE M APPING
A. Notation
An arrival process is characterized by the arrival timefunction ¯ A ( n ) , for n = 1 , , . . . , where ¯ A ( n ) denotes thearrival time of packet n . For notational convenience, we define ¯ A (0) = 0 . In addition, we define ¯ A ( m, n ) = ¯ A ( n ) − ¯ A ( m ) to be the inter-arrival time between the arrivals of packet m and packet n , for n ≥ m ≥ . For instance, ¯ A ( n, n + 1) is theinter-arrival time between packets n and n + 1 for n ≥ .As an analogy, we also characterize the arrival process usinganother function A ( t ) , t ≥ , which counts the cumulativeamount of traffic (in bits) carried by the arrival process upto time t . Similarly, we define A ( s, t ) ≡ A ( t ) − A ( s ) as thecumulative amount of traffic carried by the arrival process fromtime s to t , and for notational convenience, we let A (0) = 0 .When studying the superposition of I ( ≥ multiple arrivalprocesses, we use ¯ A i ( n ) , ( i = 1 , . . . , I ) , to denote the arrivaltime function of each individual arrival process, and ¯ A ( n ) that of the aggregate process. In addition, we use A i ( t ) , ( i = 1 , . . . , I ) , to denote the cumulative traffic amount timefunction of each individual arrival process, and A ( t ) that ofthe aggregate process. B. The TSN / DetNet Traffic Specification
The TSN / DetNet traffic specification is defined as [1][2]:
Definition 1.
An arrival process is said to conform to the TSN/ DetNet traffic specification with interval parameter τ ( > and maximum packet number parameter K ( ≥ , if during aspecified duration of length τ , the number of packets generatedby this arrival process is limited by K . For Definition 1, we have the following remarks. First,this specification aims to characterize flows at the packetlevel. We believe, there is an underlying reason for this. Inparticular, the delay of a packet at a network node is comprisedof two types of delays, namely processing related delays,and transmission related delays. Typically, delays in the firstcategory are affected only at the packet level, little by thepacket length, unlike the delays in the second category. Withthe link speed enters Gbps range, the nodal packet delaybecomes more and more dominated by the first category, forwhich packet level characterization is crucial.Second, in [1][2], there is a maximum packet length param-eter that could also be included in the TSpec. However, byconvention, the maximum packet length of a flow or arrivalprocess typically does not change in the network. For thisreason as well as the discussion above, the maximum packetlength parameter is not included in Definition 1.Third, for flows characterized by this TSpec, few results areavailable for their delay guarantee analysis. On the contrary,a rich set of such results have already been developed in NC,e.g. [3][4][5][6][7][8][9][10][11][12]. So, an idea is to find away to link TSN / DetNet TSpec to traffic models in NC,though this traffic specification is fundamentally different.In the following, we introduce a traffic model that is relatedto NC, and prove its relationship with the TSN / DetNet TSpec.
C. The Max-Plus Traffic Model and the Mapping
In this paper, we introduce the following traffic model.
Definition 2.
An arrival process is said to be ( λ, ν ) -constrained, if, for all n ≥ m ≥ , there holds ¯ A ( m, n ) ≥ λ ( n − m − ν ) + where ( x ) + ≡ max { x, } and λ ( > and ν ( ≥ are twoconstant parameters. As the definition shows, the ( λ, ν ) model is defined on thearrival time function. Indeed, it is a special case of the max-plus arrival curve model defined for the max-plus networkcalculus [5] [9] [10], where a more general function, calledmax-plus arrival curve, is used as the constraint function.The following lemma shows that, the definition of the ( λ, ν ) model is equivalent to an expression in the max-plus algebra,and is hence referred to as a max-plus traffic model . The proofis similar to that for the general max-plus arrival curve modelin Lemma 5.2 in [10] and omitted. Lemma 1.
If an arrival process is ( λ, ν ) -constrained, if andonly if, there holds ¯ A ( n ) ≥ ¯ A ¯ ⊗ ¯ α ( n ) where ¯ α ( n ) = λ ( n − m − ν ) + , and the operation ¯ ⊗ of twofunctions F ( n ) and G ( n ) is the max-plus convolution, definedas F ¯ ⊗ G ( n ) ≡ sup ≤ m ≤ n { F ( m ) + G ( n − m ) } . The following theorem establishes a relationship betweenthe TSN/DetNet TSpec and the ( λ, ν ) model. Theorem 1. (i) If an arrival process is ( λ, ν ) -constrained, itconforms to the TSN / DetNet traffic specification with (a) interval parameter τ = j/λ and maximum packet numberparameter K = ⌈ ν ⌉ + j + 1 , or (b) interval parameter τ = ( j/λ ) − and maximum packetnumber parameter K = ⌈ ν ⌉ + j ,for any integer j ≥ , where x − denotes x − ǫ for ǫ → .(ii) If an arrival process conforms to the TSN / DetNet trafficspecification with parameters τ and K ( ≥ , it is ( λ, ν ) -constrained with λ = K/τ and ν = K − .Proof. For the first part, the condition implies, for any m ≥ and for ∀ j ≥ , ¯ A ( m, m + ⌈ ν ⌉ + j + 1) ≥ ⌈ ν ⌉ + j + 1 − νλ ≥ j + 1 λ > jλ . This is to say the time distance between any two packets thatare ⌈ ν ⌉ + j +1 apart is greater than jλ . In other words, such twopackets cannot be in an interval of length jλ . Equivalently, thisis to say that in an interval of length jλ , the maximum numberof packets cannot exceed ⌈ ν ⌉ + j + 1 . Without loss of generality, suppose packet m is the first packet in theperiod. Note that from packet m to packet m + ⌈ ν ⌉ + j + 1 , there are intotal ⌈ ν ⌉ + j + 2 packets. However, since ¯ A ( m, m + ⌈ ν ⌉ + j + 1) > jλ , thelast packet, i.e. packet m + ⌈ ν ⌉ + j + 1 , cannot be within this period. So,the total number of packets in this period will not exceed ⌈ ν ⌉ + j + 1 . ndeed, for the first part, we also have ¯ A ( m, m + ⌈ ν ⌉ + j ) ≥ ¯ A ( m, m + ν + j ) ≥ jλ > (cid:18) jλ (cid:19) − . Similarly, this is to say that in an interval of length (cid:0) jλ (cid:1) − , themaximum number of packets does not exceed ⌈ ν ⌉ + j .For the second part, under the given condition, we have ¯ A ( m, n ) ≥ (cid:22) n − mK (cid:23) τ = (cid:24) n − m − K + 1 K (cid:25) Kλ − ≥ (cid:18) n − m − K + 1 K (cid:19) + Kλ − = λ − ( n − m − K + 1) + which concludes the second part. Remarks:
From the second half of Theorem 1.(i), if ν is aninteger and let j = 1 , we then obtain that if an arrival processis ( λ, ν ) -constrained, it conforms to the TSN / DetNet trafficspecification with parameters τ = (cid:0) λ (cid:1) − and K = ν + 1 .Here the mapping between ν and K in the two models is thesame as from the second part of the theorem, i.e. Theorem1.(ii). However, it is worth highlighting that for parameters λ and τ , the relation τ = (cid:0) λ (cid:1) − from the max-plus trafficmodel to the TSN / DetNet TSpec is no more recovered fromthe reverse relation from Theorem 1.(ii) where we differentlyhave λ = K/τ . This implies that the two models are in generalnot equivalent to each other.
D. The Analogy Min-Plus ( σ, ρ ) Traffic Model
The well-known ( σ, ρ ) traffic model is as the following [3]: Definition 3.
An arrival process is said to be ( σ, ρ ) -constrained, if, for all t ≥ s ≥ , A ( s, t ) ≤ ρt + σ where parameters ρ ( > and σ ( ≥ are often called the rateand burst parameters respectively. It is also known (see e.g. [6]) that the definition of the ( σ, ρ ) model is equivalent to the following, and hence referred to as a min-plus traffic model : Lemma 2.
An arrival process is ( σ, ρ ) -constrained, if andonly if, there holds A ( t ) ≤ A ⊗ α ( t ) where α ( t ) = ρt + σ , and the operation ⊗ of two func-tions F ( n ) and G ( n ) is the min-plus convolution, defined as F ⊗ G ( t ) ≡ inf ≤ s ≤ t { F ( s ) + G ( t − s ) } . Note that for any period defined by s ( ≤ t ) and t , we alwayshave A ( s, t ) = P Ii A i ( s, t ) , based on which, the superpositionproperty of the ( σ, ρ ) model is easily verified (see e.g. [6]): Lemma 3.
Consider the superposition of I ( ≥ arrivalprocesses A i ( t ) , i = 1 , . . . , I . If each arrival process A i ( t ) is ( σ i , ρ i ) -constrained, the aggregate process A ( t ) is ( σ, ρ ) -constrained with ρ = I X i =1 ρ i ; σ = I X i =1 σ i . In contrast to the min-plus ( σ, ρ ) model, for the max-plus ( λ, ν ) model, its superposition property has not been found/ proved. In fact, the superposition property of the moregeneral arrival curve model is a long-standing open problem[5][10][12]. This motivates and is focused in the next section.III. T HE S UPERPOSITION P ROPERTY OF THE M AX -P LUS T RAFFIC M ODEL
A. The Difficulty
For the superposition of arrival processes, the followingrelationship was initially derived in [10] and has also beenverified in [12]:
Lemma 4.
Given the arrival time function ¯ A i ( n ) of eachindividual process, the arrival time function ¯ A ( n ) of theaggregate process can be related to ¯ A i ( n ) as, ¯ A ( n ) = inf m + ··· + m I = n max i =1 ,...,I ¯ A i ( m i ) . (1)The expression (1) is neat, based on which, we can write ¯ A ( m, n ) = inf m + ··· + m I = n max i =1 ,...,I ¯ A i ( m i ) − inf m + ··· + m I = m max i =1 ,...,I ¯ A i ( m i ) (2)Unfortunately, it is unknown how to further relate the righthand side of (2) directly to ¯ A i ( m i , n i ) , i.e. to write theright hand side as a function of and only of ¯ A i ( m i , n i ) , i = 1 , . . . , I . This makes it difficult to find the superpositionproperty of the ( λ, ν ) model from the above relationship.To bypass this difficulty, when packet length informationis known, an indirect approach (see e.g., [5] [10]) has beenproposed. While this indirect approach is mathematicallysound, its application is limited, some compromise may haveto be made and the result can be loose. More discussion onthese will be provided in Sec. IV. B. The Superposition Property of the ( λ, ν ) Model
This subsection is devoted to finding and proving thesuperposition property of the arrival time function based ( λ, ν ) max-plus traffic model, summarized in the following theorem. Theorem 2.
Consider the superposition of I ( ≥ arrivalprocesses ¯ A i , i = 1 , . . . , I . If all arrival processes ¯ A i are ( λ i , ν i ) -constrained, the aggregate process ¯ A is ( λ dir. , ν dir. ) -constrained with λ dir. = I X i =1 λ i ; ν dir. = I X i =1 ν i + ( I − . Theorem 2 can be proved by induction. We first present thebase case with I = 2 in Lemma 5 and its proof. emma 5. Consider the superposition of two processes ¯ A i , i = 1 , . If both processes ¯ A i are ( λ i , ν i ) -constrained, theaggregate process ¯ A is ( λ, ν ) -constrained with λ = λ + λ ; ν = ν + ν + 1 . Proof.
Though lengthy, the complete proof is provided below,as we believe, the techniques used in the proof also provideinsights when dealing with similar problems. In addition, theproof itself also serves as an indication of the difficulty asdiscussed in the previous subsection.To help the presentation, we let ¯ α ( n ) = 1 λ ( n − ν ) + = 1 λ + λ ( n − ν − ν − + . Then, with the definition of the ( λ, ν ) model, to prove thelemma is to prove that, for all n ≥ m ≥ , there holds: ¯ A ( m, n ) ≥ ¯ α ( n − m ) . (3)We start with two trivial cases. One is, for any n = m ( ≥ , ¯ A ( m, n ) = 0 by definition, with which, (3) holds because ¯ α (0) = ( − ν ) + = 0 . Another is, for any n > m ( ≥ with n − m = 1 , ¯ A ( m, n ) ≥ because of non-negative inter-arrivaltime between m and m + 1 , with which, (3) holds because ¯ α (1) = ( − ν − ν ) + = 0 .Next, we consider any n > m ( ≥ with n − m > . Thecorresponding time period is [ ¯ A ( m ) , ¯ A ( n )] . We denote the setof packets between m and n in ¯ A as { m + 1 , . . . , n − } ¯ A . Without loss of generality, we suppose customer n is from ¯ A and is the n -th customer in ¯ A . In other words, we have ¯ A ( n ) = ¯ A ( n ) . (4)Under this setting, there are three possibilities about customer A ( m ) : ( Case 1 ) It is either from ¯ A , or ( Case 2 ) is from ¯ A ,or ( Case 3 ) is the virtual packet at time 0 for which we have A (0) = 0 . For the first two cases, we must have m ≥ , andfor the third case, m = 0 . Accordingly, we prove for the threecases: Case 1: Packet m in the aggregate process is from ¯ A . Let m denote its number in ¯ A , which implies: ¯ A ( m ) = ¯ A ( m ) (5) ¯ A ( m, n ) = ¯ A ( m , n ) (6)Now, given m and n are both from ¯ A , there are (and only)three sub-cases, Case 1.1 - Case 1.3, which we consider below. Case 1.1: In { m + 1 , . . . , n − } ¯ A , there is no packet from ¯ A . In this sub-case, we have: n − m = n − m . (7) This set has been intentionally used in the proof to avoid ambiguitythat would arise if the time period [ ¯ A ( m ) , ¯ A ( n )] had been used, becauseconcurrent arrivals may exist or happen both in the individual arrival processesand in the aggregate process even at ¯ A ( m ) and/or ¯ A ( n ) , which cannot bedistinguished by using [ ¯ A ( m ) , ¯ A ( n )] . In addition, since ¯ A is constrained by ( λ , ν ) , we have ( λ + λ ) · ¯ A ( m, n ) ≥ λ · ¯ A ( m, n ) = λ · ¯ A ( m , n ) ≥ ( n − m − ν ) + = ( n − m − ν ) + which gives ¯ A ( m, n ) ≥ λ + λ ( n − m − ν ) + ≥ ¯ α ( n − m ) . Case 1.2: In { m + 1 , . . . , n − } ¯ A , there is one packet from ¯ A . In this sub-case, we have: n − m = ( n − m ) + 1 (8)where, on the right hand side, the first term represents thenumber of intervals in ¯ A and the second term represents thatan additional interval is introduced because of the one packetfrom ¯ A , in { m, . . . , n } ¯ A .Similarly, we have ( λ + λ ) · ¯ A ( m, n ) ≥ λ · ¯ A ( m , n ) ≥ ( n − m ) − ν = ( n − m − − ν ) + which gives ¯ A ( m, n ) ≥ λ + λ · ( n − m − ν − + ≥ ¯ α ( n − m ) . Case 1.3: In { m +1 , . . . , n − } ¯ A , there are multiple packetsfrom ¯ A . Without loss of generality, let m be the first and n be the last of these packets from ¯ A . In this sub-case, thefollowing facts hold: ¯ A ( n ) ≥ ¯ A ( n ) (9) ¯ A ( m ) ≤ ¯ A ( m ) (10)which gives ¯ A ( m, n ) ≥ ¯ A ( m , n ) . In addition, we have n − m = ( n − m ) + ( n − m ) + 1 (11)where the left hand side represents the number intervalsbetween packets m and n in ¯ A . For the right hand side, in { m, . . . , n } ¯ A , we now have ( n − m + 1) packets from ¯ A ,and ( n − m + 1) packets from ¯ A , which in total gives ( n − m ) + ( n − m ) + 2 ≡ N number of packets that have N − intervals, which is ( n − m ) + ( n − m ) + 1 .We then have ( λ + λ ) · ¯ A ( m, n )= λ · ¯ A ( m , n ) + λ · ¯ A ( m, n ) ≥ λ · ¯ A ( m , n ) + λ · ¯ A ( m , n ) ≥ ( n − m − ν ) + + ( n − m − ν ) + ≥ (( n − m − ν ) + ( n − m − ν )) + = (( n − m − − ( ν + ν )) + (12)and hence ¯ A ( m, n ) ≥ λ + λ ( n − m − ν − ν − + = ¯ α ( n − m ) . ombing Case 1.1 - Case 1.3, (3) is proved for the firstcase. In the following, we consider the second case. Case 2: Packet m in the aggregate process is from ¯ A . Without of generality, suppose it is the m -th packet in ¯ A ,which also implies ¯ A ( m ) = ¯ A ( m ) (13)In this case, there are also (and only) three sub-cases, Case2.1 - Case 2.3, which we consider below. Case 2.1: In { m + 1 , . . . , n − } ¯ A , there is no packet from ¯ A but there is at least one packet from ¯ A . Let n denotethe last such packet from ¯ A . Based on the definition of n ,we must have ¯ A ( n ) ≤ ¯ A ( n ) = ¯ A ( n ) (14) n − m = ( n − m ) + 1 (15)where, on the right hand side of (15), the first term ( n − m ) represents the number of intervals of packets from ¯ A and thesecond term represents the additional interval introduced bythe one packet, i.e. n , from ¯ A in { m, . . . , n } ¯ A .With (13) and (14), we now have, ( λ + λ ) · ¯ A ( m, n ) ≥ λ · ( ¯ A ( n ) − ¯ A ( m )) ≥ λ · ( ¯ A ( n ) − ¯ A ( m )) ≥ ( n − m − ν ) + = ( n − m − − ν ) + and hence ¯ A ( m, n ) ≥ λ + λ ( n − m − ν − + ≥ ¯ α ( n − m ) . Case 2.2: In { m + 1 , . . . , n − } ¯ A , there is no packet from ¯ A but there is at least one packet from ¯ A . Let m denotethe first such packet from ¯ A . Based on the definition of m ,we must have ¯ A ( m ) ≥ ¯ A ( m ) = ¯ A ( m ) (16) n − m = ( n − m ) + 1 (17)where, on the right hand side of (17), the first term ( n − m ) represents the number of intervals of packets from ¯ A and thesecond term represents that an additional interval is introducedby the one packet, i.e. m , from ¯ A in { m, . . . , n } ¯ A .With (4), (16) and (17), we now have, ( λ + λ ) · ¯ A ( m, n ) ≥ λ · ( ¯ A ( n ) − ¯ A ( m )) ≥ λ · ( ¯ A ( n ) − ¯ A ( m )) ≥ ( n − m − ν ) + = ( n − m − − ν ) + and hence ¯ A ( m, n ) ≥ λ + λ ( n − m − ν − + ≥ ¯ α ( n − m ) . Case 2.3: In { m + 1 , . . . , n − } ¯ A , there is at least onepacket from ¯ A and there is at least one packet from ¯ A . Let m denote the first such packet from ¯ A , and n the last such packet from ¯ A . Based on the definitions of m and n , wemust have ¯ A ( m ) ≥ ¯ A ( m ) = ¯ A ( m ) (18) ¯ A ( n ) ≤ ¯ A ( n ) = ¯ A ( n ) (19) n − m = ( n − m ) + ( n − m ) + 1 (20)where (18) is the same as (16), (19) the same as (14), and onthe right hand side of (20), the first term ( n − m ) representsthe number of intervals of packets from ¯ A , the second term ( n − m ) represents the number of intervals of packest from ¯ A , and the third term represents that an additional intervalneeds to be added due to the superposition, all in { m, . . . , n } ¯ A .(See also the discussion for (11).)With (18) and (19), we now have, ( λ + λ ) · ¯ A ( m, n )= λ · ( ¯ A ( n ) − ¯ A ( m )) + λ · ( ¯ A ( n ) − ¯ A ( m )) ≥ λ · ( ¯ A ( n ) − ¯ A ( m )) + λ · ( ¯ A ( n ) − ¯ A ( m )) ≥ ( n − m − ν ) + + ( n − m − ν ) + ≥ (( n − m ) − ν + ( n − m ) − ν ) + = ( n − m − − ν − ν ) + (21)and hence ¯ A ( m, n ) ≥ λ + λ ( n − m − ν − ν − + = ¯ α ( n − m ) . Combing Case 2.1 - Case 2.3, (3) is proved for the secondcase. With this, we have proved (3) holds for all n > m ≥ . Case 3: Customer m is the virtual packet at the origin,i.e. m = 0 and A (0) = 0 . In this case, in addition to the n customers from ¯ A , there are n − n customers from ¯ A inthe period, and we must also have ¯ A ( n − n ) ≤ ¯ A ( n ) = ¯ A ( n ) with which, we further obtain ( λ + λ ) · ¯ A (0 , n )= λ · ¯ A ( n ) + λ · ¯ A ( n ) ≥ λ ¯ A ( n ) + λ · ¯ A ( n − n ) ≥ ( n − ν ) + + (( n − n ) − ν ) + ≥ (( n − ν ) + (( n − n ) − ν )) + = ( n − ν − ν ) + and hence ¯ A (0 , n ) ≥ λ + λ (( n − − ν − ν ) + ≥ ¯ α ( n − . This, together with the proof for Case 1 and Case 2, ends theproof of Lemma 5.Next for the induction, we prove Theorem 2 also holdsfor I + 1 arrival processes, given the condition that it holdsfor I arrival processes. Note that, under the given condition,the aggregate process of I arrival processes is ( λ ( I ) , ν ( I ) ) -constrained with λ ( I ) = P Ii =1 λ i ; ν ( I ) = P Ii =1 ν i + ( I − . The aggregate process of I + 1 arrival processes, denoted as ¯ A ( I +1) , can be treated as the superposition of two processes A ( I ) and ¯ A I +1 , where ¯ A ( I ) denotes the aggregate of the first I processes and ¯ A I +1 the last process. Then, with Lemma 5, ¯ A ( I +1) is ( λ ( I +1) , ν ( I +1) ) -constrained with λ ( I +1) = λ ( I ) + λ I +1 = I +1 X i =1 λ i ν ( I +1) = ν ( I ) + ν I +1 + 1 = I +1 X i =1 ν i + (( I + 1) − which is Theorem 2 for the superposition of I + 1 processes.This completes the proof of Theorem 2 . C. Extensions
It is worth highlighting that the superposition property ofthe ( λ, ν ) model presented in Theorem 2 resembles that of the ( σ, ρ ) model shown in Lemma 3.Following the essence in the proof of Theorem 2, thefollowing superposition property can be proved for the TSN /DetNet traffic specification. Corollary 1.
Consider the superposition of I ( ≥ arrivalprocesses ¯ A i , i = 1 , . . . , I . If each arrival process ¯ A i confirmsto the TSN / DetNet traffic specification with interval τ i andmaximum packet number K i , then the aggregate process ¯ A also confirms to the TSN / DetNet traffic specification withinterval τ and maximum packet number K where τ − = I X i =1 τ − i ; K = I X i =1 K i . In addition, the superposition property of the ( λ, ν ) modelcan be extended to the more general max-plus arrival curvemodel shown below. Corollary 2.
Consider the superposition of I ( ≥ arrivalprocesses ¯ A i , i = 1 , . . . , I . If each of them has a max-plus arrival curve ¯ α i ( · )( ≥ , ( i = 1 , . . . , I ) , then thesuperposition process ¯ A has a max-plus arrival curve ¯ α as ¯ α ( n ) = 1 λ · ( n − ν ) + where λ = I X i =1 λ i ; ν = I X i =1 ν i + ( I − with λ i = sup { r : r · ¯ α i ( n ) ≤ n } (22) ν i = n − λ i ¯ α i ( n ) . (23) Proof.
Under the given assumptions, we can re-write ¯ α i ( n ) as ¯ α i ( n ) = 1 λ i ( n − ν i ) + . Then, the result follows immediately from Theorem 2 .Furthermore, it is worth highlighting that, with the help ofTheorem 1 and Theorem 2, the existing NC results can bemade use of for delay guarantee analysis of TSN / DetNet. With (22), i.e. the definition of λ i , ν i is non-negative in nature. IV. C
OMPARISON
In this section, we compare the superposition results ob-tained using the indirect approach and those using the directapproach proposed in the literature (see e.g., [5] [10]).Specifically, the indirect approach first transforms the ( λ, ν ) characterization from the arrival time function to the ( σ, ρ ) traffic characterization, then applies the superposition propertyof the ( σ, ρ ) model to find the ( σ, ρ ) characterization for theaggregate process, and finally transforms the obtained ( σ, ρ ) characterization back to the the ( λ, ν ) characterization.The following lemma summarizes the result from the indi-rect approach. Its proof is omitted, since a general but muchmore complex form can be found from Corollary 6.2.9 in [5]. Lemma 6.
Consider the superposition of I ( ≥ arrivalprocesses ¯ A i , i = 1 , . . . , I . If each ¯ A i is ( λ i , ν i ) -constrainedwith maximum packet length l i and the minimum packet lengthof all processes is known, denoted as l , then the aggregateprocess ¯ A is ( λ ind. , ν ind. ) -constrained with λ ind. = I X i =1 l i l λ i ; ν ind. = I X i =1 ( ν i + 1) l i l . Comparing Lemma 6 with Theorem 2, in addition to howtheir results are derived, there are two fundamental differences : • For Lemma 6 to be applicable, we at least need to knowthe maximum packet length of each process and theminimum packet length of all processes. In contrast, nospecific packet length information is required for Theo-rem 2. This difference has an immediate consequence,which is, if the packet length information is not knownor provided, the superposition result presented in Lemma6 can no more be used. • Even when the needed packet length information con-dition for Lemma 6 to be applicable is available, itsresultant ( λ, ν ) representation is worse than what isfrom Theorem 2. This is because λ ind. ≥ λ dir. and ν ind. > ν dir. leading to a smaller or worse boundingfunction λ − ( n − m − ν ) in the ( λ, ν ) characterization.In the rest, we presents results for four extremely simplecases to exemplify the comparison. For simplicity in the ex-pression, we assume every flow i produces packets periodicallyand the period length is τ i . In addition, for ease of expression,we consider the superposition of only two flows, i.e. I = 2 .The other settings of the four cases are: • Case 1: All flows have the same period τ i = τ . • Case 2: All flows have the same period τ i = τ and thesame packet length l i = l . • Case 3: All flows still have the same packet length l i = l ,but while one flow has period τ = τ , the other flow hasperiod τ = 2 τ . • Case 4: All other settings are the same as for the secondcase, except that the second flow has packet length l =2 l . As a remark, in this case, the average traffic rate (inbps) of the second flow is the same as that of the firstflow, i.e. ρ = ρ = l/τ . ABLE IC
OMPARISON OF SUPERPOSITION PROPERTY RESULTS
Cases: Indirect Appr. (Lemma 6) Direct Appr. (Theorem 2)Case 1 Not Available τ ( n − + Case 2 τ ( n − + τ ( n − + Case 3 τ ( n − + 2 τ ( n − + Case 4 τ ( n − + 2 τ ( n − + Table I summarizes and compares the superposition resultsfrom both approaches for the four cases. Though simple, thecomparison validates the discussion about the fundamentaldifferences between the indirect and direct approaches.V. C
ONCLUSION
The emerging time-sensitive networking (TSN) and deter-ministic networking (DetNet) standards (re-)call attention tothe network calculus, in order to make use of the rich set ofresults available in NC. In this paper, we introduced an arrivaltime function based max-plus NC traffic model. We provedthat it is closely related to the TSN TSpec and there is a directmapping between them. In addition, another focus has been onfinding and proving the superposition property of the max-plustraffic model, providing answer to a long-standing question inthe max-plus network calculus. The proof adopted a noveldirect approach that requires no packet length information,in contrast to a literature indirect approach. Appealingly,the proved superposition property shows clear analogy withthat of the well-known counterpart ( σ, ρ ) model in NC. Thecomparison of the superposition results from the indirect anddirect approaches not only shows wider applicability of thesuperposition property obtained in this paper, but also offersbetter traffic characterization for the aggregate process. Theseresults can help make use of the NC results for delay guaranteeanalysis of TSN / DetNet networks.A CKNOWLEDGMENT
This is an updated version. The initial version of this paperwas submitted to IEEE Globecom 2018 and will be presentedthere. The author would like to thank its anonymous reviewersfor their helpful comments, and Jean-Yves Le Boudec forsimilar comments. It is mainly based on those commentsthat this updated version has been produced. In addition, theauthor would like to specially thank Jean-Yves Le Boudecfor pointing out that there is an equivalent model of the ( λ, ν ) model, which is called “packet burstiness” constraintPB ( ρ, K ) independently introduced in [13], and that based onthe PB model and results in [13], a simplified proof of thesuperposition property for the ( λ, ν ) model may be obtained.R EFERENCES[1] Time-Sensitive Networking Task Group of IEEE 802.1. IEEEP802.1Qcc/D1.6. July 18, 2017.[2] N. Finn, P. Thubert, B. Varga, and J. Farkas. Deterministic networkingarchitecture. draft-ietf-detnet-architecture-04 , October 30, 2017.[3] R. L. Cruz. A calculus for network delay, part I: network elements inisolation.
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